An Introduction to Modal Logic V

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1 An Introduction to Modal Logic V Axiomatic Extensions and Classes of Frames Marco Cerami Palacký University in Olomouc Department of Computer Science Olomouc, Czech Republic Olomouc, November 7 th 2013 Marco Cerami (UPOL) Modal Logic V / 20

2 Introduction Introduction we have proved that the minimal normal modal logic K is complete with respect to the class of all Kripke frames; we can do the same with some particular axiomatic extensions of K; in this case we consider particular classes of frames; these axiomatic extensions have usually been already considered in the literature before of being studied in the framework of Kripke semantics; the classes of frames are defined by particular properties of the accessibility relation; by means of the methods of canonical models, it is possible to prove a correspondence between axiomatic extensions and classes of frames. Marco Cerami (UPOL) Modal Logic V / 20

3 Introduction Axiomatic Extensions of K Axiomatic Extensions of K Marco Cerami (UPOL) Modal Logic V / 20

4 Introduction Axiomatic Extensions of K The Logic K Definition A normal modal logic Λ is a set of formulas containing: all classical tautologies (in the modal language), (p q) ( p q) (axiom (K)), p p, p p, and is closed under: (MP) Modus Ponens: if ϕ Λ and ϕ ψ Λ, then ψ Λ, (US) Uniform Substitution: if ϕ Λ, then ψ Λ, where ψ is obtained from ϕ by replacing propositional variables by arbitrary formulas, (G) Generalization: if ϕ Λ, then ϕ Λ. Marco Cerami (UPOL) Modal Logic V / 20

5 Introduction Axiomatic Extensions of K Axiomatic Extensions: axioms (4) p p (T) (B) (D) (E) (M) (G) (L) p p p p p p p p p p p p ( p p) p Marco Cerami (UPOL) Modal Logic V / 20

6 Introduction Axiomatic Extensions of K Axiomatic Extensions: logics T KT K4 K4 S4 KT 4 B KTB S5 KT 4B or KT 4E GL KL D KD D4 KD4 Marco Cerami (UPOL) Modal Logic V / 20

7 Introduction Classes of Frames Classes of Frames Marco Cerami (UPOL) Modal Logic V / 20

8 Kripke frames Introduction Classes of Frames A Kripke frame is a structure F = W, R, where: W is a non-empty set of elements, often called possible worlds, R W W is a binary relation on W, called the accessibility relation of W. d e f g h i c b a Marco Cerami (UPOL) Modal Logic V / 20

9 Classes of frames Introduction Classes of Frames R is reflexive ( x)r(x, x) R is symmetric ( x y)(r(x, y) R(y, x)) R is transitive ( x y z)(r(x, y) R(y, z) R(x, z)) R is euclidean ( x y z)(r(x, y) R(x, z) R(y, z)) R is serial ( x y)r(x, y) R is a partial function ( x y z)(r(x, y) R(x, z) y = z) R is a function R is a function and is serial R is dense ( x y)(r(x, y) ( z)(r(x, z) R(z, y))) R is Noetherian there are no infinite ascending R-chains Marco Cerami (UPOL) Modal Logic V / 20

10 Frame Completeness and Canonical Frames Marco Cerami (UPOL) Modal Logic V / 20

11 Frame Completeness We say that a formula ϕ is a (local) consequence in a class of frames F of a set of formulas Σ if for every point w of every model M on every frame F F it happens that in symbols: M, w Σ M, w ϕ Σ F ϕ For any normal modal logic Λ, we say that it is sound with respect to a class of frames F if, for any set of formulas Σ ϕ: Σ Λ ϕ Σ F ϕ For any normal modal logic Λ, we say that it is (strongly) complete with respect to a class of frames F if, for any set of formulas Σ ϕ: Σ F ϕ Σ Λ ϕ Marco Cerami (UPOL) Modal Logic V / 20

12 Remarks the soundness proof consists in proving that the axiom that extends K is sound in the class of frames considered; about completeness, consider that: Σ F ϕ Σ Λ ϕ if and only if Σ Λ ϕ Σ F ϕ So, we have to find a frame F F such that F Σ but F ϕ, under the assumption that ϕ is not deducible from Σ; if Λ is complete with respect to its canonical frame F Λ, it is enough to prove that F Λ F. Marco Cerami (UPOL) Modal Logic V / 20

13 T and the class of reflexive frames Example I: the logic T and the class of reflexive frames Marco Cerami (UPOL) Modal Logic V / 20

14 T and the class of reflexive frames T and the class of reflexive frames T is the axiomatic extension of K by means of axiom: (T ) : p p About soundness: we have already proved that the duality axioms and (K) are sound in every frame; in the same way we have proved that deductive rules (MP), (US) and (G) preserve the truth in every frame; we only need to prove that (T ) is sound in every reflexive frame. About completeness: we need to build a reflexive frame which, under the supposition that Σ T ϕ, satisfies Σ but not ϕ; we only need to prove that the canonical frame FT is reflexive. Marco Cerami (UPOL) Modal Logic V / 20

15 Soundness T and the class of reflexive frames Let F = W, R be a frame and assume that R is reflexive, let M = W, R, V be any model on F and w W, suppose that M, w p, hence ( v)(r(w, v) V (p, v)), then p is true in every point v such that R(w, v), since R is reflexive, we have that R(w, w), then p is true at point w, so, p p is true at w. Marco Cerami (UPOL) Modal Logic V / 20

16 T and the class of reflexive frames Completeness Assume that p p is a valid formula of Λ, then p p, for every maximally Λ-consistent set, hence p p is true at every point of the canonical frame F Λ = W Λ, R Λ ; let W Λ and let ϕ, by (MP), we have that ϕ, by definition of R Λ, we have that R(, ) iff {ψ : ψ } hence R(, ), so, F Λ = W Λ, R Λ is reflexive. Marco Cerami (UPOL) Modal Logic V / 20

17 K4 and the class of transitive frames Example II: the logic K4 and the class of transitive frames Marco Cerami (UPOL) Modal Logic V / 20

18 K4 and the class of transitive frames K4 and the class of transitive frames K4 is the axiomatic extension of K by means of axiom: (4) : p p About soundness: we have already proved that the duality axioms and (K) are sound in every frame; in the same way we have proved that deductive rules (MP), (US) and (G) preserve the truth in every frame; we only need to prove that (4) is sound in every transitive frame. About completeness: we need to build a transitive frame which, under the supposition that Σ T ϕ, satisfies Σ but not ϕ; we only need to prove that the canonical frame F4 is transitive. Marco Cerami (UPOL) Modal Logic V / 20

19 K4 and the class of transitive frames Soundness Let F = W, R be a frame and assume that R is transitive, let M = W, R, V be any model on F and w W, suppose that M, w p, hence ( v)(r(w, v) V (p, v)), then p is true in every point x such that R(w, x), let v, u W be any points such that R(w, v) and R(v, u), by transitivity we have that R(w, u) hence p is true at point u, then p is true at point v, therefore p is true at point w, so, p p is true at w. Marco Cerami (UPOL) Modal Logic V / 20

20 Completeness K4 and the class of transitive frames Assume that p p is a valid formula of Λ, then p p, for every maximally Λ-consistent set, hence p p is true at every point of the canonical frame F Λ = W Λ, R Λ ; let,, W Λ be such that R Λ (, ) and R Λ (, ), if ϕ, by (MP), we have that ϕ, by definition of R Λ, we have that R(, Γ) iff {ψ : ψ } Γ hence ϕ and ϕ, therefore R Λ (, ), so, F Λ = W Λ, R Λ is transitive. Marco Cerami (UPOL) Modal Logic V / 20

An Introduction to Modal Logic III

An Introduction to Modal Logic III Soundness of Normal Modal Logics Marco Cerami Palacký University in Olomouc Department of Computer Science Olomouc, Czech Republic Olomouc, October 24 th 2013 Marco Cerami